The present invention is directed to a system for developing number sense and arithmetic skills, and specifically to a ten-frame subtraction system for teaching subtraction skills.
The present invention builds on a ten-frame dot card system developed by the inventor of the present invention. The ten-frame dot card system includes ten-frame cards 100 (hereinafter “cards”) that show a positive numerical representation of the numbers 0–10 in two-by-two arrays of “dots” framed in ten windows. As shown in FIG. 1, each card 100 has a combination of “dots” 102 and “empties” 104. The number of dots 102 shown on the card 100 is the number that the card 100 represents. Because the dots 102 and empties 104 are framed in ten (10) windows, each number is shown in relation to ten (10) it helps to reinforce concepts of place value, necessary for basic arithmetic. These cards 100 are used to model numbers for place value and computation. The system sometimes includes a carrier that has pockets for each number 0–10 in which the cards 100 are generally stored.
FIG. 1 shows an exemplary embodiment of the ten-frame dot card system. Although only one set of cards 100 is shown, for each number 0–10 (each having the appropriate number of dots 102) there may be multiple cards 100. For example, in one preferred embodiment the ten-frame dot card system would include two (2) copies of each card 100 representing 0–9 and ten (10) copies of the card 100 representing 10, for a total of thirty (30) cards 100. The cards 100 representing the number 10 may be a different color than the cards 100 representing the numbers 0–9. If the system includes a carrier, multiple cards 100 representing the same number could be stored in the pocket imprinted with the corresponding numeral.
The term “card 100” is used throughout this specification to describe a device upon which dots 102 may be imprinted, marked, etched, or otherwise represented. It should be noted that the cards 100 might be plastic or heavy paper cards, plastic tiles, coins, chips, pieces of paper, pages in a book, or any other device that can be imprinted or marked.
The term “dot 102” is used throughout this specification to describe graphical representations or markings on the cards 100, each dot 102 representing one numerical unit. It should be noted that the dots 102 may be replaced with other graphical representations or markings such as lines, hatch marks, stars, smiley faces, flowers, pictures of animals or characters, letters or numbers, or any other marking similarly meant to designate one numerical unit.
The term “window” is used throughout this specification to describe an empty space. Although the shown embodiments have lines outlining the windows, these lines are optional.
The ten-frame dot card system shown in FIG. 1 was shown and described in the article, Developing Number Sense and Arithmetic Skills with Ten-Frame Dot Cards written by the inventor of the present invention as well as other publications by the inventor. The cards 100 used in the ten-frame dot card system could be used for many activities including putting numbers in order and number flash (in which children quickly count the dots 102 as they are shown a “flash”). Other activities are described in Success with Math Coach publications (also authored by the inventor of the present invention) including Ten-Frame Games and Activities, Dot Card Tutorial, Add and Subtract Volume 1: Basic Facts, Ten-Frame Activities for Arithmetic Readiness, and Place Value Volume 1: Whole Numbers. These publications were primarily focused on counting and addition.
An example of how the ten-frame dot card system was used to teach subtraction is shown in Add and Subtract Volume 1: Basic Facts. The method for teaching subtraction was a write-out method that involved crossing out dots 102 on the card 100 to “take away” the designated number of dots 102. The number of dots 102 left represents the solution to the equation. A first example of how this works is the equation “5−2=3.” For this first equation the user would choose a card 100 with five (5) dots 102 and then cross out two (2) dots 102. The three (3) dots 102 remaining uncrossed would represent the solution to the equation which, in this case would be “3.” A second example of how this works is the equation “6−6=0.” For this second equation the user would choose a card 100 with six (6) dots 102 and then cross out six (6) dots 102. The zero (0) dots 102 remaining uncrossed would represent the solution to the equation which, in this case would be “0.” A third example of how this works is the equation “8−0=8.” For this third equation the user would choose a card 100 with eight (8) dots 102 and then cross out zero (0) dots 102. The eight (8) dots 102 remaining uncrossed would represent the solution to the equation which, in this case would be “8.”
There were a variety of problems with the write-out method. First, it involves using a writing instrument. A user must first obtain a writing instrument which he may not have readily available. It is also easy to loose a writing instrument. For younger users, the parent or teacher may also want to avoid the use of writing instruments which can be distracting and messy. If the writing instrument is permanent in nature (e.g. the writing cannot be removed once it is on the card), the cards will not be suitable for reuse. Even pencil markings tend to be impossible to completely erased. The use of non-permanent markers is also problematic. First, the cards must be coated or covered with a wipable surface. Second, it requires special non-permanent markers that tend to be expensive. Third, non-permanent markers tend to dry out. Fourth, the use of a non-permanent marker requires an eraser or cloth, which is another part to lose. Fifth, the ink on the surface of a card that has been written on tends to smear easily on hands of the user as well as any other item or surface it touch touches. Sixth, for some students, the write-out method makes it impossible for them to review all parts of the equation because they can see only the total or the result at any given time.
Another method of teaching subtraction using “Totally Ten Grids” (2×5 grids) is described in Every Day Counts by Patsy F. Kanter. In this reference blue and red dot stickers are used to populate the grids. One of the exercises is set forth as follows: “When there are more than five dots in the Grid, ask children to tell how many blue and red dots make the total. If you ask how many are left if we take away the blue ones and how many are left if we take away the red ones, some children may come to see that if you know 5 and 2 are 7, then 7 take away 2 will always leave the 5.” Subtraction is taught, therefore, using the different colored dots. There were a variety of problems with this method. First, it involves using a bunch of little dot stickers. If the user runs out of dot stickers, the product is no longer useful. It is also easy to loose little dot stickers. For younger users, the parent or teacher may also want to avoid the use of little dot stickers which can be distracting as the children stick them all over everything except the grids. If the dot stickers are permanent in nature (e.g. the stickers cannot be removed once they are on the card), the cards will not be suitable for reuse. Further, the arrangement of the dots does not encourage children to learn other ways to subtract from 7 such as 7−3, 7−4, 7−1, and 7−6. Finally, dot color coding limits the educational value of this model.